Module Details |
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module. |
Title | Geometry of Continued Fractions | ||
Code | MATH447 | ||
Coordinator |
Dr O Karpenkov Mathematical Sciences O.Karpenkov@liverpool.ac.uk |
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Year | CATS Level | Semester | CATS Value |
Session 2024-25 | Level 7 FHEQ | Second Semester | 15 |
Aims |
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To give an introduction to the current state of the art in geometry of continued fractions and to study how classical theorems can be visualized via modern techniques of integer geometry. |
Learning Outcomes |
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(LO1) Determine best approximations to real numbers and to homogeneous decomposable forms. |
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(LO2) Apply the techniques of geometric continued fractions to quadratic irrationalities (Lagrange’s theorem, Markov spectrum). |
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(LO3) Apply methods of lattice trigonometry in the study of toric varieties. |
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(LO4) Compute relative frequencies of faces in multidimensional continued fractions. |
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(LO5) Apply the methods of multidimensional continued fractions to study properties of algebraic irrationalities of higher degree. |
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(S1) Problem solving skills |
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(S2) Numeracy |
Syllabus |
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Introduction to theory of continued fractions: generic continued fractions, approximation properties. Lattice geometry, including Pick’s theorem, integer invariants in terms of group indices. Lattice trigonometry. Basic relations, classification of integer triangles. Toric surfaces, Ikea problem for toric singularities. Quadratic irrationalities. Lagrange's theorem on the periodicity of continued fractions, the algorithm of Gauss Reduction. Basics of ergodic theory. Gauss-Kuzmin theorem on the distribution of elements of continued fractions. Multidimensional continued fractions in the sense of Klein, White’s theorem, classification of two-dimensional faces. Generalized Lagrange theorem. Multidimensional Gauss-Kuzmin distribution. Farey tessellation. |
Recommended Texts |
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Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module. |
Pre-requisites before taking this module (other modules and/or general educational/academic requirements): |
MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH102 CALCULUS II; MATH244 Linear Algebra and Geometry |
Co-requisite modules: |
Modules for which this module is a pre-requisite: |
Programme(s) (including Year of Study) to which this module is available on a required basis: |
Programme(s) (including Year of Study) to which this module is available on an optional basis: |
Assessment |
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EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
written exam | 90 | 50 | ||||
CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
Homework 3 | 0 | 10 | ||||
Homework 2 | 0 | 10 | ||||
Homework 1 | 0 | 10 | ||||
Homework 5 | 0 | 10 | ||||
Homework 4 | 0 | 10 |